FYS Vår 2016 (Kondenserte fasers fysikk)

Transcription

1 FYS Vår 2016 (Kondenserte fasers fysikk) Pensum: Introduction to Solid State Physics by Charles Kittel (Chapters 1-9 and 17, 18, 20) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen 23a

2

3 Lecture 4: Atomic diffusion Phenomenology of diffusion: describing diffusion in terms of diffusion flux Microscopic diffusion mechanisms Sb diffusion as a function of temperature as stress in Si/SiGe heterostructures

4 Lecture 4: Atomic diffusion Phenomenology of diffusion: describing diffusion in terms of diffusion flux Microscopic diffusion mechanisms Sb diffusion as a function of temperature as stress in Si/SiGe heterostructures

5 Diffusion Phenomenon of material transport by atomic or particle transport from region of high to low concentration What forces the particles to go from left to right? Does each particle know its local concentration? Every particle is equally likely to go left or right! At the interfaces in the above picture, there are more particles going right than left this causes an average flux of particles to the right! Largely determined by probability & statistics

6 Diffusion Glass tube filled with water. At time t = 0, add some drops of ink to one end of the tube. Measure the diffusion distance, x, over some time.

7 Describing diffusion in terms of diffusion flux Flux: amount of material or atoms moving past a unit area in unit time Flux, J = DM/(A Dt) Directional Quantity Flux can be measured for: --vacancies --host (A) atoms --impurity (B) atoms

8 Describing diffusion in terms of diffusion flux Concentration Profile, C(x): [kg/m 3 ] Cu flux Ni flux Concentration of Cu [kg/m 3 ] Concentration of Ni [kg/m 3 ] Fick's First Law: Position, x

9 Describing diffusion in terms of diffusion flux Steady State: Steady rate of diffusion from one end to the other. Implies that the concentration profile doesn't change with time. Why? Apply Fick's First Law: If J x ) left = J x ) right, then J x D dc dx dc dc dx left dx right Result: the slope, dc/dx, must be constant (i.e., slope doesn't vary with position)!

10 Describing diffusion in terms of diffusion flux Concentration profile, C(x), changes w/ time. To conserve matter: Fick's First Law: Governing Eqn.: Fick s second law

11 Lecture 4: Atomic diffusion Phenomenology of diffusion: describing diffusion in terms of diffusion flux Microscopic diffusion mechanisms Sb diffusion as a function of temperature as stress in Si/SiGe heterostructures

12 Type of point defects vacancy Interstitial impurity Substitutional impurity

13 Diffusion mechanisms Diffusion at the atomic level is a step-wise migration of atoms from lattice site to lattice site Conditions for diffusion: there must be an adjacent empty site atom must have sufficient energy to break bonds with its neighbors and migrate to adjacent site ( activation energy) Higher the temperature, higher is the probability that an atom will have sufficient energy hence, diffusion rates increase with temperature Types of atomic diffusion mechanisms: substitutional (through vacancies) interstitial

14 Diffusion mechanisms Substitutional Diffusion: applies to substitutional impurities atoms exchange with vacancies rate depends on: -- number of vacancies -- temperature -- activation energy to exchange.

15 Diffusion mechanisms Initial state Intermediate state Final state Energy Activation energy Also called energy barrier for diffusion

16 Lecture 4: Atomic diffusion Phenomenology of diffusion: describing diffusion in terms of diffusion flux Microscopic diffusion mechanisms Sb diffusion as a function of temperature as stress in Si/SiGe heterostructures

17

18

19 Strained silicon How does it work? Basic idea: Change the lattice constant of material

20 Zn self-diffusion revealing V Zn energetics - astonishing progress is reached calculating intrinsic defect energetics in ZnO by ab initio methods, however the experimental verification is missing from A.Janotti and C.G. Van de Walle, Phys.Rev. B (2007) - Zn self-diffusion as method assessing intrinsic defect energetics D V Zn D V 0 e ( E f VZn E m VZn )/ kt D I Zn I ( E D0 e f ZnI E m ZnI )/ kt

21 Defect formation energy (ev) Zn self-diffusion revealing V Zn energetics from A.Janotti and C.G. Van de Walle, Phys.Rev. B (2007) E f ZnI : Zn rich (i) experimentally accessible E F values are in the upper part of the band gap (ii) theoretically predicted trends for V Zn and Zn I formation energies go in opposite directions E f VZn : O rich E f VZn : Zn rich (iii) Zn self-diffusion activation energy is a combination of the formation and migration energies of defects mediating the diffusion

22 Defect formation energy (ev) Zn self-diffusion revealing V Zn energetics from A.Janotti and C.G. Van de Walle, Phys.Rev. B (2007) E f ZnI : Zn rich (i) experimentally accessible E F values are in the upper part of the band gap (ii) theoretically predicted trends for V Zn and Zn I formation energies go in opposite directions E f VZn : O rich E f VZn : Zn rich (iii) Zn self-diffusion activation energy is a combination of the formation and migration energies of defects mediating the diffusion Thus, the trend for Zn diffusion activation energy should follow the defect formation energy trends for V Zn or Zn I assuming the defect migtration energy is independent of the Fermi level and the chemical potential. D V Zn D V 0 e ( E f VZn E m VZn )/ kt D I Zn I ( E D0 e f ZnI E m ZnI )/ kt

23 Zn self-diffusion revealing V Zn energetics SIMS Intensity intensity nat ZnO 64 ZnO nm MgO buffer C-orriented Al 2 O 3 substrate Samples - isotopic nat ZnO/ 64 ZnO heterostructures grown by rf-mbe sample Z2 64Zn 66Zn 68Zn 70Zn Depth (nm)

24 Zn self-diffusion revealing V Zn energetics Concentration (cm -3 ) Concentration (cm -3 ) nat ZnO 64 ZnO C-orriented Al 2 O 3 substrate Samples - isotopic nat ZnO/ 64 ZnO heterostructures grown by rf-mbe Sample-Z Zn 69 Ga Depth (nm) Sample-Z Ga 70 Zn Depth (nm)

25 Zn self-diffusion revealing V Zn energetics 70 Zn concentration (cm -3 ) 70 Zn concentration (cm -3 ) o C, 30 min (a) (b) o C, 30 min Depth (nm) Zn-rich samples as-grown Z1 Z2 Z3 simulations Depth (nm) Z1 sample as-grown annealed O1 sample as-grown annealed simulations

26 Zn self-diffusion revealing V Zn energetics 70 Zn concentration (cm -3 ) 70 Zn concentration (cm -3 ) o C, 30 min (a) (b) o C, 30 min Depth (nm) Zn-rich samples as-grown Z1 Z2 Z3 simulations Depth (nm) Z1 sample as-grown annealed O1 sample as-grown annealed simulations

27 Zn self-diffusion revealing V Zn energetics Diffusivity (cm 2 /s) Temperature ( o C) sample E a Zn Z Z Z3 2.6 O /T (K -1 )

28 Zn self-diffusion revealing V Zn energetics Electron concentration (cm -3 ) e concentration (10 18 cm -3 ) E F (ev) 19 Z1 10 Z2 Z3 O E n i F E N i F c nc ktln, ni N v 1/ 2 exp E g T 2kT, Temperature ( o C) 10 N c, v 2 2m h e, h 2 kt 3 / 2, /-0.03 ev E C Temperature ( o C) 1.5

29 V Zn formation energy (ev) E a of Zn self-diffusion (ev) Zn self-diffusion revealing V Zn energetics 6 4 Zn-rich from A.Janotti and C.G. Van de Walle, Phys.Rev. B (2007) Z1 Z2 Z3 O1 O O-rich ev Fermi level (ev) 0

30 V Zn formation energy (ev) E a of Zn self-diffusion (ev) Zn self-diffusion revealing V Zn energetics 6 4 Zn-rich from A.Janotti and C.G. Van de Walle, Phys.Rev. B (2007) Z1 Z2 Z3 O1 O O-rich Fermi level (ev) 1.4 ev E a values follow the V Zn energy formation trend suggesting Zn to diffuse in ZnO via vacancy mechanism, i.e E Zn a E f VZn E m VZn

31 V Zn formation energy (ev) E a of Zn self-diffusion (ev) Zn self-diffusion revealing V Zn energetics 6 4 Zn-rich from A.Janotti and C.G. Van de Walle, Phys.Rev. B (2007) Z1 Z2 Z3 O1 O O-rich Fermi level (ev) 1.4 ev Accounting that Zn f m E and the upper limit of the V a EVZn EVZn Zn migration energy can not be higher than the lowest E a value, we can substract 1.4 ev to obtain estimates of V Zn formation energies. 0

32 Zn self-diffusion revealing V Zn energetics V Zn formation energy (ev) 4 3 Z1 Z2 Z3 O1 Zn-rich 2 1 O-rich Fermi level (ev)